On absolute colimits

An absolute characterisation of locally determined omega-colimits

Characterising colimiting ω-cocones of projection pairs in terms of least upper bounds of their embeddings and projections is important to the solution of recursive domain equations. We present a universal characterisation of this local property as ω-cocontinuity of locally continuous functors. We present a straightforward proof using the enriched Yoneda embedding. The proof can be generalised ...

On colimits and elementary embeddings

We give a sharper version of a theorem of Rosický, Trnková and Adámek [12], and a new proof of a theorem of Rosický [13], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)extendible cardinals.

A Primer on Homotopy Colimits

On Sifted Colimits and Generalized Varieties

Filtered colimits, i.e., colimits over schemes D such that D-colimits in Set commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that D-colimits in Set commute with finite products. An important example: reflexive coequalizers are sifted colimits. Generalized varieties are defined as free completions of small categories under sift...

On Colimits in Various Categories of Manifolds

Homotopy theorists have traditionally studied topological spaces and used methods which rely heavily on computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what i...